Integration using method of substitution

Problem:
(tdt)/(4-t^4)^(1/2)
Attempt:
I want the derivate of whatever i make u equal to, to equal something outside of u therefore I will factorize the denominator to equal -(-2+t^2)(2+t^2) and make u equal to (2+t^2) so that du=2tdt
Balance the equation so that one side is equal to the numerator
(1/2)du=tdt
Now I have
1/2[tex]\int[/tex] = 1/(-(-2+t^2)u)^1/2
How do I apply this elementary integral in this situation?:
[tex]\int[/tex] 1/(a^2-x^2)^1/2 dx = sin^-1(x/a) + C
I don't know what to do with the u in
1/(-(-2+t^2)u)^1/2